In many engineering applications, the assessment of reliability has to be done within a limited amount of information, which does not allow to use exact values for the distributional hyperparameters. This is achieved defining probability boxes and assessing the reliability computing the failure probability bounds. Probability boxes are often obtained from known probability distribution functions represented by interval hyper-parameters. In the applications, not only it is of interest estimating the failure probability bounds, but it is also required to identify the extreme realizations leading to the estimated bounds. In this paper, we propose a strategy, based on the Kolmogorov-Smirnov test, to identify the parental distribution function that best fit the distribution of extreme realizations, obtained from the minmax propagation. From the results obtained comparing the strategy with a direct search, it has emerged that the proposed method is generally applicable and efficient.